What is the next approach for chronic sinusitis/cough if antibiotics/steroids/nasal rinse/adenoidectomy doesn't fix it ?

Possibilities...: You could add a steroid nasal spray to more aggressively treat the sinusitis. You could also add a treatment for gerd if the cough does not completely resolve. Good luck!

Lifestyle audit: Discovering every irritant (smoke/fume/fragrace/particulate) that may aggravate your sx & working to avoid them.(quit smoking) using electrostatic ac filters & changing them regularly. Using advanced particle filters in your sleeping quarters.Using a crossover irritant/allergy med like singulair (montelukast). Having an allergy eval & undergoing immune-therapy.